To determine the performance or behavior of a power system under unbalanced conditions it is essential to know sequence impedance and network of power system elements. This is a piece of fundamental knowledge for performing the unsymmetrical fault analysis in the power system. Here we will be discussing the sequence impedance and networks of various power system elements.** **

**Sequence Impedance and Network of Power System Elements: Synchronous Machine**

For determining the sequence impedance and network of power system elements, let us consider an unloaded synchronous machine which can be a generator or motor. It is grounded through a reactor of impedance Z_{n}. E_{a}, E_{b,} and E_{c} are the EMFs induce in the three phases of the machine respectively.

An unsymmetrical fault on the machine terminals causes unbalanced current I_{a}, I_{b,} and Ic to flow in the respective phases. If this fault involves the ground then-current Inflows to neutral from the ground through reactor Z_{n}. The current I_{n} is the phasor sum of I_{a}, I_{b,} and I_{c}.

These unbalanced currents can be resolved into symmetrical currents and respective positive, negative, and zero sequence networks of the machine can be formed.

**Positive Sequence Impedance and Network of Synchronous Machine**

Synchronous machines offer time-varying reactance X_{d}”(Sub-transient reactance), X_{d}‘(transient reactance) and X_{d}(synchronous reactance).

For a circuit where the sudden value of current under switching or fault condition is to be determined, the Sub-transient reactance of the machine is used. To determine the current after the few-cycle of switching or fault (3 or 4 cycles), transient reactance X_{d}‘ is used. For steady-state conditions, synchronous reactance X_{d} is used.

Positive sequence impedance along with the reactance of the machine makes up positive sequence sub transient, transient, or steady-state impedance of the machine respectively.

Given is the positive sequence network of synchronous machine

The positive sequence impedance of the synchronous machine is represented by a source emf on no load and the positive sequence impedance of the machine. The phasor sum of positive sequence currents of the three phases I_{a1}, I_{b1}, I_{c1} are zero, so no current flows through neutral and Zn doesn’t appear in the network.

**Negative Sequence Impedance and Network of Synchronous Machine**

Negative sequence reactance X2 of the synchronous machine oscillates between the direct axis reactance X_{d}”and quadrature axis reactance X_{q}”. X2 value is usually taken average and given by

**Zero Sequence Impedance and Network of Synchronous Machine**

The current flowing in the neutral through reactor impedance Z_{n} is the sum of the zero-sequence currents in all three phases. So, the voltage drop caused by this sum of zero sequence current is 3I_{a0}Z_{n}. The voltage drop of the zero-sequence terminal is 3I_{a0}Z_{n }+ I_{a0}Z_{g0}. The zero sequence impedance is given by Z_{g0.}

The zero-sequence network of the transmission line is shown below.

Zero sequence voltage of terminal ‘a’ is

**Sequence Impedance and Network of Power System Elements: Transmission Line**

For a transmission line, positive and negative sequence impedances are equal. Zero sequence impedance includes the impedance of the return path through the ground is different from positive and negative sequence impedance. Zero sequence impedance of transmission line is much larger than the positive and negative sequence impedance.

For a fully transposed transmission line, there is no mutual coupling between the sequence networks.

**Sequence Impedance and Network of Loads**

A star-connected load with isolated neutral has no path for the flow of zero sequence current. It offers infinite impedance between the neutral and ground leaving behind the zero-sequence network open-circuited between the neutral of the star-connected load and the reference bus.

When the neutral of the star-connected load is grounded through a reactor of impedance Z_{n} then zero-sequence voltage drop will occur due to the flow of zero sequence current through Z_{n} given by 3I_{a0} Z_{n}. This is the same as that of I_{a0} flowing through 3Z_{n}. So, an impedance of 3Z_{n} is introduced between the neutral terminal and the reference bus of the zero-sequence network as shown.

For a balanced load, the positive, negative, and zero-sequence impedance of the load are equal.

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